The schematic illustration of the proposed microscope is shown in Fig. 1a, which consists of four main components: an EDOF and zoom microscope objective, an image sensor, an illumination system, and an image processing system. As an essential element, the varifocal lens is specially designed and manufactured to help the system achieve not only continuous optical zoom with a large zoom ratio, but axial scanning with constant magnification. Combined with the proposed image fusion algorithm, the proposed microscope can perform two functions: EDOF and optical zoom. For EDOF function, the proposed microscope can realize EDOF with constant magnification and high resolution. By varying the curvature of the varifocal lenses, a series of 2D images with the same magnification are obtained, which are then rapidly processed by the proposed image algorithm to generate the EDOF image. as shown in Figure 1b. The optical zoom function is shown in Fig. 1 C. The combination of varifocal lenses is optimized to achieve different magnifications without any mechanical movement. Thus, the proposed microscope can obtain the EDOF images at any magnification in real time without mechanical movement, which is not available for existing microscopes.

As a key part of the proposed microscope, the EDOF and zoom microscope objective consists of several glass lenses and four varifocal lenses. A simplified diagram of the microscope objective is shown in Fig. 2a. Four varifocal lenses provide four variables to achieve optical axial scanning, zooming and maintaining constant magnification with high resolution. In this way, mechanical movement of the sample during scanning and zooming can be avoided. For conventional optical sectioning techniques, the magnification of the acquired image varies with scan depth, and it is difficult to correct for the problem of inconsistent magnification by using a single focal length as the variable. However, the introduction of varifocal lenses can increase the degree of freedom of the microscope objective, which can be used to correct the problem of inconsistency when extending the DOF and achieving optical zoom.

In order to take full advantage of the zoom capability of the vari-focal lens, we adopt a front and rear group splicing design in the optical path. The image plane of the front group coincides with the object plane of the rear group, and the image space NA of the front group corresponds to the object space NA of the rear group. This way we can use the front group to increase the NA of the lens, while the rear group is used to extend the zoom ratio. Additionally, the front and rear groups have zoom capability, so they can work together to achieve continuous optical zoom by controlling the curvature of all four varifocal lenses. In addition, the change in magnification (M_{{text{Front}}}) of the front group is caused by the change of the object distance and the change of the magnification (M_{{{text{Back}}}}) of the rear group is caused by the change in the distance of the image. The optical distance of the front and rear groups remains unchanged, so the zoom magnification equation can be obtained as follows,

$$ frac{{1 – M_{Before}^{2} }}{{M_{Before}^{2} }}f_{Before}^{prime } dM_{Before} + frac{{1 – M_{{{text{Re}} ar}}^{2} }}{{M_{{{text{Re}} ar}}^{2} }}f_{{{text{Re}} ar}}^{prime } dM_{{{text{Re}} ar}} = 0, $$

(1)

where (M_{{{text{Front}}}}), (M_{{{text{Back}}}}) are magnifications of the front group and rear group of the proposed lens, respectively. Outraged, (f_{{{text{Front}}}}^{prime }), (f_{{{text{Back}}}}^{prime }) are the focal length of the lens, the front group and the back group, respectively.

The cross-sectional magnification of the microscope objective ((M_{{{text{Ob}}}})) can be given by the equation below.

$$ M_{{{text{Ob}}}} = M_{{{text{Front}}}} * M_{{{text{Back}}}} , $$

(2)

Considering the characteristics of the front and rear groups in the light path, we use the two varifocal lenses of the front group as the focusing group to extend the EDOF of the microscope and maintain constant magnification. As shown in Fig. 2b, by adjusting the curvature of the two zoom lenses, the microscope objective quickly focuses to a certain depth, extending the DOF of the microscope from (z_{0}) at (z_{1}). Since there is only one variable curvature surface in the varifocal lens, we can simplify it to a simple plano-convex lens for ease of calculation. To keep the proposed objective magnification constant, the focal length of two zoom lenses must satisfy the equation. (3).

$$ frac{{f_{P11}^{prime } }}{{f_{P21}^{prime } }} = frac{{z_{1} – f_{G}^{prime } } }{{z_{0} – f_{G}^{prime } }} * frac{{f_{P12}^{prime } }}{{f_{P22}^{prime } }}, $ $

(3)

where (f{^{prime}}_{P11}), (f{^{prime}}_{P21}), (f{^{prime}}_{P12}) and (f{^{prime}}_{P22}) are the corresponding focal lengths of varifocal lens 1 and lens 2 when the focusing distances of the lens are z0 and z1, respectively. (f{^{prime}}_{G}) is the focal length of the glass lens group.

According to the above theoretical analysis, the proposed lens can realize continuous optical zoom and extend DOF with constant magnification based on specially designed parameters.